3.821 \(\int \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q x \, dx\)

Optimal. Leaf size=98 \[ -\frac{b \left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right )}{2 a^2 (p+1)} \]

[Out]

-(b*(a + b/x^2)^(1 + p)*(c + d/x^2)^q*AppellF1[1 + p, -q, 2, 2 + p, -((d*(a + b/
x^2))/(b*c - a*d)), (a + b/x^2)/a])/(2*a^2*(1 + p)*((b*(c + d/x^2))/(b*c - a*d))
^q)

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Rubi [A]  time = 0.213183, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{b \left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,2;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right )}{2 a^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^p*(c + d/x^2)^q*x,x]

[Out]

-(b*(a + b/x^2)^(1 + p)*(c + d/x^2)^q*AppellF1[1 + p, -q, 2, 2 + p, -((d*(a + b/
x^2))/(b*c - a*d)), (a + b/x^2)/a])/(2*a^2*(1 + p)*((b*(c + d/x^2))/(b*c - a*d))
^q)

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Rubi in Sympy [A]  time = 27.2869, size = 76, normalized size = 0.78 \[ - \frac{b \left (\frac{b \left (- c - \frac{d}{x^{2}}\right )}{a d - b c}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p + 1} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,2,- q,p + 2,\frac{a + \frac{b}{x^{2}}}{a},\frac{d \left (a + \frac{b}{x^{2}}\right )}{a d - b c} \right )}}{2 a^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**p*(c+d/x**2)**q*x,x)

[Out]

-b*(b*(-c - d/x**2)/(a*d - b*c))**(-q)*(a + b/x**2)**(p + 1)*(c + d/x**2)**q*app
ellf1(p + 1, 2, -q, p + 2, (a + b/x**2)/a, d*(a + b/x**2)/(a*d - b*c))/(2*a**2*(
p + 1))

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Mathematica [B]  time = 0.699803, size = 229, normalized size = 2.34 \[ \frac{b d x^2 (p+q-2) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{2 (p+q-1) \left (x^2 \left (a d p F_1\left (-p-q+2;1-p,-q;-p-q+3;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+2;-p,1-q;-p-q+3;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )-b d (p+q-2) F_1\left (-p-q+1;-p,-q;-p-q+2;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b/x^2)^p*(c + d/x^2)^q*x,x]

[Out]

(b*d*(-2 + p + q)*(a + b/x^2)^p*(c + d/x^2)^q*x^2*AppellF1[1 - p - q, -p, -q, 2
- p - q, -((a*x^2)/b), -((c*x^2)/d)])/(2*(-1 + p + q)*(-(b*d*(-2 + p + q)*Appell
F1[1 - p - q, -p, -q, 2 - p - q, -((a*x^2)/b), -((c*x^2)/d)]) + x^2*(a*d*p*Appel
lF1[2 - p - q, 1 - p, -q, 3 - p - q, -((a*x^2)/b), -((c*x^2)/d)] + b*c*q*AppellF
1[2 - p - q, -p, 1 - q, 3 - p - q, -((a*x^2)/b), -((c*x^2)/d)])))

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q}x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^p*(c+d/x^2)^q*x,x)

[Out]

int((a+b/x^2)^p*(c+d/x^2)^q*x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q*x,x, algorithm="maxima")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (x \left (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\right )^{q}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q*x,x, algorithm="fricas")

[Out]

integral(x*((a*x^2 + b)/x^2)^p*((c*x^2 + d)/x^2)^q, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**p*(c+d/x**2)**q*x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^p*(c + d/x^2)^q*x,x, algorithm="giac")

[Out]

integrate((a + b/x^2)^p*(c + d/x^2)^q*x, x)